For multiple applications in the field of radars, it is desirable to filter a digitized signal with a specific transfer function.
In order to do this, use is made of transversal-type finite impulse response filters. Such filters are often referred to by the acronym FIR that is derived from the term “Finite Impulse Response” as per the accepted English terminology. FIR filters effectively implement operations that include the use of time shifts of the signal, of gains and summation. The number of operations is equal to the length of the impulse response of the FIR filter being considered (with the length being expressed in number of samples).
However, when the length of the impulse response of the filter is very large, as is the case for the pulse compression that is involved in the radars, the process of carrying out the filtering becomes problematic or even impossible in view of the very large number of operations involved.
In order to work around such a problem, it is a known practice to perform some operation in the space of frequencies. For this, a Fourier transformation is applied in order to pass from the domain of time to the domain of frequencies, with the filter operation thus then becoming multiplicative, then a Fourier transformation is subsequently applied so as to return into the domain of time.
In practice, the time is divided into sequences and the Fourier transformation is effectively implemented by means of a fast Fourier transform often referred to by the abbreviation FFT for the accepted term “Fast Fourier Transform”. More precisely, the passage from the domain of the time space to the frequency space is obtained through the use of a FFT while the passage from the domain of the frequency space to the time space is obtained through the use of an IFFT. The abbreviation IFFT refers to the term “Inverse Fast Fourier Transform” as per the accepted terminology.
The use of Fourier Transforms, whether inverse fast FT or not, entails a size that is at least equal to the length of the filter. In fact, if the size of the Fourier transform, whether inverse fast FT or not, is strictly equal to the size K of the filter, then the process makes it possible to obtain only one single point on K, with the K−1 other points calculated not being usable. If the size of the Fourier Transform, whether inverse fast FT or not, is strictly equal to two times the size K of the filter, that is to say 2K, then the process makes it possible to obtain K points on 2K, with the K other points calculated not being usable. Thus, by doubling the process, it is possible to calculate 2 times K points on 2K and to access all of the required points.
However, this shows that half of the points calculated will be lost, which increases the computational load and complicates the operational implementation of the filter.